Binary kernel matrices of maximum exponents of polar codes of dimensions up to sixteen

The pioneering work of Arikan on polar codes is based on a kernel matrix of dimension two and exponent 0.5. To achieve larger exponent in order to improve performance, kernel matrices of larger dimensions are considered. In this paper, constructions of binary kernel matrices of dimensions up to 16 with maximum exponents are presented. The results show that the minimum dimension for which there exists a kernel matrix with exponent greater than 0.5, i.e., exceeds the exponent of the kernel matrix proposed by Arikan, is 15.